Numerische Mathematik

, Volume 121, Issue 4, pp 637–670 | Cite as

A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

Article

Abstract

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.

Mathematics Subject Classification (2000)

65M12 82D37 

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques UMR 6620, CNRSUniversité Blaise PascalAubière CedexFrance

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